Tsitsipas was trailing 4-2 in the first set when he bowed out citing a longstanding issue to his right arm. The result means Popyrin will advance to the next round, where he will face compatriot James Duckworth.
"I haven't retired once in my life, and it was something that I had to do today. ... I'm trying to be cautious for the next tournament, which is the most important one for me. I have had an issue there for quite a while now," Tsitsipas said after the match.
"It has gotten bigger in the past couple of weeks, so I'm just trying to protect it. I felt the pain playing in the match, and I just don't want for it to get worse than it is now.
"I know how to treat it, but playing every day, going out on the court practicing doesn't make it better.
"I'm going to have treatment. I'm going to try and have the best people treat me and give me the best advice -- anything possible to recover and be 100% again."
Tsitsipas, who has won two Tour titles this season as well as finishing second at the French Open, is next scheduled to play in the ATP Finals, which begins Nov. 14.
The second-seeded Russian squandered three set points at 5-1 in the first set and then allowed Ivashka to rally for 5-5. However, Ivashka then dropped serve with a double fault and three unforced errors, and Medvedev clinched the set with a backhand volley.
The US Open champion broke for a 2-1 lead in the second but then hit double-faults on his first two match points before converting the third when Ivashka sent a forehand wide.
Novak Djokovic, who is also playing in Paris, has a lead of 800 points over Medvedev in the rankings and could clinch the year-end No. 1 this week depending on their respective results.
Alcaraz, who won when they last faced off in Spain in 2019, claimed a straight sets 7-6 (1), 7-5 victory over Sinner on Tuesday, denting the Italian's hopes at qualifying for the ATP Finals.
"I'm so happy for this win as Jannik was fighting for a spot at the [Nitto] ATP Finals," Alcaraz said after the match. "It's my third Top 10 win of the year. I think Jannik and I will have a great rivalry in the future."
Norrie, 26, won his first Masters title at Indian Wells last month, and another triumph in Paris this week would see the Brit qualify for one of the two remaining ATP Finals spots in Turin.
Poland's Hubert Hurkacz is leading the race for Finals qualifying as he sits in the eighth spot after also reaching the third round in Paris, with a win over American Tommy Paul on Wednesday. To date, Norway's Casper Ruud holds the other qualifying place.
In other results, fourth-seeded Alexander Zverev advanced, as did Gael Monfils and Grigor Dimitrov, while Felix Auger-Aliassime and Pablo Carreno Busta were both ousted. Americans Marcos Giron and Sebastian Korda pulled upsets, beating Diego Schwartzman and Marin Cilic, respectively.
Information from The Associated Press was used in this report.